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<div class="section boost_icl_concepts_sets_and_maps" lang="en">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_icl.concepts.sets_and_maps"></a><a class="link" href="sets_and_maps.html" title="Sets and Maps">Sets and Maps</a>
</h3></div></div></div>
<a name="boost_icl.concepts.sets_and_maps.a_set_concept"></a><h6>
<a name="id1076180"></a>
        <a class="link" href="sets_and_maps.html#boost_icl.concepts.sets_and_maps.a_set_concept">A Set Concept</a>
      </h6>
<p>
        On the fundamental aspect all <code class="computeroutput"><a class="link" href="../../boost/icl/interval_base_set.html" title="Class template interval_base_set">interval_sets</a></code>
        are models of a concept <code class="computeroutput"><span class="identifier">Set</span></code>.
        The <code class="computeroutput"><span class="identifier">Set</span></code> concept of the Interval
        Template Library refers to the mathematical notion of a set.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              <p>
                Function
              </p>
              </th>
<th>
              <p>
                Variant
              </p>
              </th>
<th>
              <p>
                implemented as
              </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
              <p>
                empty set
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span><span class="special">::</span><span class="identifier">Set</span><span class="special">()</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                subset relation
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="keyword">bool</span> <span class="identifier">Set</span><span class="special">::</span><span class="identifier">within</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s1</span><span class="special">,</span>
                <span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s2</span><span class="special">)</span><span class="keyword">const</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                equality
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="keyword">bool</span> <span class="identifier">is_element_equal</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s1</span><span class="special">,</span>
                <span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                set union
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">+=</span>
                <span class="special">(</span><span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span> <span class="keyword">operator</span>
                <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                set difference
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">-=</span>
                <span class="special">(</span><span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span> <span class="keyword">operator</span>
                <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                set intersection
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">&amp;=</span>
                <span class="special">(</span><span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Set</span> <span class="keyword">operator</span>
                <span class="special">&amp;</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Set</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Set</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        Equality on <code class="computeroutput"><span class="identifier">Sets</span></code> is not implemented
        as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>,
        because <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>
        is used for the stronger lexicographical equality on segments, that takes
        the segmentation of elements into account.
      </p>
<p>
        Being models of concept <code class="computeroutput"><span class="identifier">Set</span></code>,
        <a href="http://www.cplusplus.com/reference/stl/set/" target="_top"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">set</span></code>
        </a> and all <code class="computeroutput"><a class="link" href="../../boost/icl/interval_base_set.html" title="Class template interval_base_set">interval_sets</a></code>
        implement these operations and obey the associated laws on <code class="computeroutput"><span class="identifier">Sets</span></code>. See e.g. <a href="http://en.wikipedia.org/wiki/Algebra_of_sets" target="_top">an
        algebra of sets here</a>.
      </p>
<a name="boost_icl.concepts.sets_and_maps.making_intervals_complete"></a><h6>
<a name="id1077070"></a>
        <a class="link" href="sets_and_maps.html#boost_icl.concepts.sets_and_maps.making_intervals_complete">Making
        intervals complete</a>
      </h6>
<p>
        An <code class="computeroutput"><a class="link" href="../../boost/icl/interval.html" title="Struct template interval">interval</a></code> is considered
        to be a set of elements as well. With respect to the <code class="computeroutput"><span class="identifier">Set</span></code>
        concept presented above <code class="computeroutput"><a class="link" href="../../boost/icl/interval.html" title="Struct template interval">interval</a></code>
        implements the concept only partially. The reason for that is that addition
        and subtraction can not be defined on <code class="computeroutput"><a class="link" href="../../boost/icl/interval.html" title="Struct template interval">intervals</a></code>.
        Two intervals <code class="computeroutput"><span class="special">[</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">]</span></code>
        and <code class="computeroutput"><span class="special">[</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">]</span></code>
        are not addable to a <span class="emphasis"><em><span class="bold"><strong>single</strong></span></em></span>
        new <code class="computeroutput"><a class="link" href="../../boost/icl/interval.html" title="Struct template interval">interval</a></code>. In other
        words <code class="computeroutput"><a class="link" href="../../boost/icl/interval.html" title="Struct template interval">intervals</a></code> are incomplete
        w.r.t. union and difference. <code class="computeroutput"><a class="link" href="../../boost/icl/interval_set.html" title="Class template interval_set">Interval_sets</a></code>
        can be defined as the <span class="emphasis"><em><span class="bold"><strong>completion</strong></span></em></span>
        of intervals for the union and difference operations.
      </p>
<p>
        When we claim that addition or subtraction can not be defined on intervals,
        we are not considering things like e.g. interval arithmetics, where these
        operations can be defined, but with a different semantics.
      </p>
<a name="boost_icl.concepts.sets_and_maps.a_map_concept"></a><h6>
<a name="id1077199"></a>
        <a class="link" href="sets_and_maps.html#boost_icl.concepts.sets_and_maps.a_map_concept">A Map Concept</a>
      </h6>
<p>
        On the fundamental aspect <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code>
        and all <code class="computeroutput"><a class="link" href="../../boost/icl/interval_base_map.html" title="Class template interval_base_map">interval_maps</a></code>
        are models of a concept <code class="computeroutput"><span class="identifier">Map</span></code>.
        Since a <code class="computeroutput"><span class="identifier">map</span></code> is a <code class="computeroutput"><span class="identifier">set</span> <span class="identifier">of</span> <span class="identifier">pairs</span></code>, we try to design the <code class="computeroutput"><span class="identifier">Map</span></code> concept in accordance to the <code class="computeroutput"><span class="identifier">Set</span></code> concept above.
      </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              <p>
                Function
              </p>
              </th>
<th>
              <p>
                Variant
              </p>
              </th>
<th>
              <p>
                implemented as
              </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
              <p>
                empty map
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span><span class="special">::</span><span class="identifier">Map</span><span class="special">()</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                subset relation
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="keyword">bool</span> <span class="identifier">within</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">,</span>
                <span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s2</span><span class="special">)</span><span class="keyword">const</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                equality
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="keyword">bool</span> <span class="identifier">is_element_equal</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s1</span><span class="special">,</span>
                <span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                map union
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">+=</span>
                <span class="special">(</span><span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span> <span class="keyword">operator</span>
                <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                map difference
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">-=</span>
                <span class="special">(</span><span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span> <span class="keyword">operator</span>
                <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
                map intersection
              </p>
              </td>
<td>
              <p>
                inplace
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="keyword">operator</span> <span class="special">&amp;=</span>
                <span class="special">(</span><span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
<tr>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
              </p>
              </td>
<td>
              <p>
                <code class="computeroutput"><span class="identifier">Map</span> <span class="keyword">operator</span>
                <span class="special">&amp;</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">Map</span><span class="special">&amp;</span> <span class="identifier">s1</span><span class="special">,</span> <span class="keyword">const</span>
                <span class="identifier">Map</span><span class="special">&amp;</span>
                <span class="identifier">s2</span><span class="special">)</span></code>
              </p>
              </td>
</tr>
</tbody>
</table></div>
<p>
        As one can see, on the abstract kernel the signatures of the icl's <code class="computeroutput"><span class="identifier">Set</span></code> and <code class="computeroutput"><span class="identifier">Map</span></code>
        concepts are identical, except for the typename. While signatures are identical
        The sets of valid laws are different, which will be described in more detail
        in the sections on the <a class="link" href="../semantics/sets.html" title="Sets">semantics
        of icl Sets</a> and <a class="link" href="../semantics/maps.html" title="Maps">Maps</a>.
        These semantic differences are mainly based on the implementation of the
        pivotal member functions <code class="computeroutput"><span class="identifier">add</span></code>
        and <code class="computeroutput"><span class="identifier">subtract</span></code> for elements
        and intervals that again serve for implementing <code class="computeroutput"><span class="keyword">operator</span>
        <span class="special">+=</span></code> and <code class="computeroutput"><span class="keyword">operator</span>
        <span class="special">-=</span></code>.
      </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2007 -2010 Joachim Faulhaber<br>Copyright &#169; 1999 -2006 Cortex Software GmbH<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
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